Every Separable Complex Fréchet Space with a Continuous Norm Is Isomorphic to a Space of Holomorphic Functions

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We start with the following observation. If a Fréchet space F is isomorphic to a Fréchet space E ⊂ H(G) of holomorphic functions on an open connected domain G ⊂ C, then F has a continuous norm. Indeed, let T : F → E be a topological isomorphism and let K ⊂ G be an infinite compact set. Since both T and the inclusion E ⊂ H(G) are continuous by assumption, there is a seminorm p on F such that supz∈K |T (x)(z)|≤ p(x) for each x ∈ F . If p(x) = 0 for some x ∈ F , then T (x)(z) = 0 for each z ∈ K. Therefore, the holomorphic function T (x) vanishes on G. Since T is injective, x = 0.

We need the following Lemma to prove our main result.

Lemma 2.1 Let F be a separable, infinite dimensional Fréchet space with a continuous ′ ′ norm. Then one can find two sequences (en)n∈N0 ⊂ F and (en)n∈N0 ⊂ F such that

(i) span{en | n ∈ N0} is dense in F . ′ ′ (ii) (en)n∈N0 is equicontinuous in F . ′ ′ ′ (iii) span{en | n ∈ N0} is dense in (F ,σ(F F )). ′ ′ N ′ (iv) hen, emi = 0 if n 6= m and hen, eni = 1 for each n ∈ 0. That is, (en, en)n∈N0 is a biorthogonal system.

Proof. We denote by τ the metrizable topology of the Fr´echet space F and by p the continuous norm on F . Let U := {x ∈ F | p(x) ≤ 1} be the unit ball of the norm p. The polar set U ◦ of U in F ′ coincides with {u ∈ F ′ | |u(x)| ≤ p(x) for all x ∈ F } and, by ◦ ◦ ′ ′ Hahn-Banach theorem, its linear span H := span(U )= ∪s∈NsU is σ(F , F )-dense in F .

Since (F,τ) is separable, there is a countable, inﬁnite subset A = (yn)n∈N0 of F which is τ dense in F . Hence, A is also dense in the normed space (F,p), and this normed space is separable, too. The topological dual of (F,p) coincides with H = span(U ◦) ⊂ F ′. Since (F,p) is metrizable and separable, it follows from [14, Corollary 2.5.13] that H is σ(H, F )- ′ separable. We select a countable, inﬁnite subset A = (vn)n∈N0 of H which is dense in

(H,σ(H, F )). We now apply [14, Proposition 2.3.2] to ﬁnd sequences (xn)n∈N0 ⊂ span(A) ′ ′ ′ ′ and (xn)n∈N0 ⊂ span(A ) such that hxn,xmi = 0 if n 6= m and hxn,xni = 1 for each N N ′ N ′ n ∈ 0, and span({xn,n ∈ 0}) = span(A) and span({xn,n ∈ 0}) = span(A ). N ′ ′ ◦ N Since for each n ∈ 0 we have xn ∈ span(A ) ⊂ H = ∪s∈NsU , then for each n ∈ 0 we ′ ′ −1 ′ can select m(n) > 0 such that |xn(x)|≤ m(n)p(x) for each x ∈ F . We set en := m(n) xn and en := m(n)xn for each n ∈ N0 and we check that these sequences satisfy properties (i) − (iv). (i) span({en,n ∈ N0}) = span({xn,n ∈ N0}) = span(A) is dense in (F,τ). ′ N ′ (ii) |en(x)|≤ p(x) for each x ∈ F and each n ∈ 0. Therefore, (en)n∈N0 is equicontinuous in F ′. ′ N ′ N ′ (iii) span({en,n ∈ 0}) = span({xn,n ∈ 0}) = span(A ) is σ(H, F )-dense in H. Since ′ ′ N ′ ′ H is σ(F , F )-dense in F , we conclude that span{en,n ∈ 0} is σ(F , F )-dense in F . ′ (iv) This follows easily from the deﬁnitions, since (xn,xn)n∈N0 is a biorthogonal system. ✷

Lemma 2.1 is a version for separable Fr´echet spaces with a continuous norm of [6, Lemma 2], which was relevant in linear dynamics. See also [1, Lemma 2.11]. Observe that ′ ′ the existence of a sequence (en)n∈N0 ⊂ F satisfying (ii) and (iii) in Lemma 2.1 implies that the space F has a continuous norm. In fact, by (ii) there is a seminorm p on F such

2 ′ that supn∈N0 |en(x)| ≤ p(x) for all x ∈ F . Suppose that p(y) = 0 for some y ∈ F . Then ′ N en(y) = 0 for each n ∈ 0. Condition (iii) implies that y = 0. The statement and proof of our next result are inspired by Mashreghi and Ransford [11, Theorem 1.3].

Theorem 2.2 Let F be a separable, infinite dimensional, complex Fréchet space with a continuous norm. Let G be an open connected domain in C. Then there exists a Fréchet space E ⊂ H(G) of holomorphic functions on G such that F is isomorphic to E and the polynomials are dense in E.

Proof. We apply Lemma 2.1 to the space F to select the sequences (en)n∈N0 ⊂ F and ′ ′ (en)n∈N0 ⊂ F satisfying conditions (i)−(iv). In particular, we find a continuous norm p on ′ N F such that |en(x)|≤ p(x) for each x ∈ F and each n ∈ 0. Now take a sequence (αn)n∈N0 ∞ n of positive numbers such that Pn=0 αnk < ∞ for each k ≥ 1. Define T : F → H(G) ∞ ′ n by T (x)(z) := Pn=0 αnen(x)z for each x ∈ F and each z ∈ G. The operator T is well- defined, linear and continuous. First of all, the series defining T (x) converges and defines an entire function for each x ∈ F , since ∞ ∞ ′ n n X αn|en(x)||z| ≤ p(x) X αnk < ∞ n=0 n=0 for each |z| ≤ k and each k ≥ 1. The continuity of T can be seen as follows: given an arbitrary compact subset L of G, there is k ≥ 1 such that |z| ≤ k for each z ∈ L. Then, for each x ∈ F we get ∞ ∞ ′ n n sup |T (x)(z)| = sup X αnen(x)z ≤ X αnk p(x). ∈ ∈ z L z L n=0 n=0 ′ Moreover, the map T is injective. Indeed, if T (x)=0in H(G), then en(x) = 0 for each ′ ′ ′ n ∈ N0. Since span({en | n ∈ N0}) is dense in (F ,σ(F , F )), this implies that x = 0. −1 n The monomials are clearly contained in T (F ), because T ((αn) en) = z for each N s n s −1 n ∈ 0. Therefore, each polynomial P (z) = Pn=0 anz = T (Pn=0 an(αn) en) belongs also to T (F ). The linear map T : F → T (F ) ⊂ H(G) is a continuous bijection. We endow E := T (F ) with the metrizable, complete topology such that T : F → E is an isomorphism. Then E is a separable Fréchet space of holomorphic functions on the domain G. The polynomials are contained in E and they are also dense. This can be seen in the following way. By (i) −1 in Lemma 2.1, span({(αn) en,n ∈ N0}) = span({en | n ∈ N0}) is dense in F . Then the set P of all polynomials satisfies

−1 −1 P = T (span({(αn) en),n ∈ N0})) = T (span({(αn) en,n ∈ N0})) = T (F )= E. ✷

A Fréchet space F has the approximation property if there is a net (Tα)α of finite rank operators on F such that Tαx converges to x for each x ∈ F uniformly on the compact subsets of F . A Fréchet space F has the bounded approximation property if there is an equicontinuous net (Tα)α of finite rank operators on F such that Tαx converges to x for each x ∈ F . If F has the bounded approximation property, then it has the approximation property, since pointwise convergence and uniform convergence on compact sets coincide on the equicontinuous subsets of the space L(F ) of all continuous linear operators from F into itself; see e.g. [10, Theorem 39.4(2)]. Assume now that F is also

3 separable. In this case, every equicontinuous subset of the space L(F ) is metrizable for the topology of pointwise convergence by [10, Theorem 39.5(9)]. Therefore, one can apply the uniform boundedness principle to conclude that a separable Fréchet space F has the bounded approximation property if and only if there is a sequence (Tn)n of finite rank operators such that limn→∞ Tnx = x for each x ∈ F . Every Fréchet space with a Schauder basis has the bounded approximation property and every nuclear Fréchet space has the approximation property. The problem of Grothendieck whether every nuclear Fréchet space has the bounded approximation property was open for quite a while. The first counterexample was due to Dubinsky, and simpler examples were obtained by Vogt. We refer the reader to the introduction of Vogt’s paper [17] for more information. In that paper Vogt presented an easy and transparent example of a nuclear Fréchet space failing the bounded approximation property and consisting of C∞-functions on a subset of R3. The first examples of nuclear Fréchet spaces with the bounded approximation property without basis are due to Mitiagin and Zobin; see pages 514 and ff. in [9]. An easy example of a nuclear Fréchet space which consists of C∞-functions and has no Schauder basis was also given by Vogt in [16]. Theorem 2.2 permits us to obtain in an abstract way examples among nuclear Fréchet spaces of holomorphic functions.

Corollary 2.3 There are nuclear Fréchet spaces of holomorphic functions on the unit disc or the complex plane without the bounded approximation property, and there are others with the bounded approximation property without Schauder basis. It is also a consequence of Theorem 2.2 that classical counterexamples in the theory of Fréchet spaces exist in the frame of separable Fréchet spaces of holomorphic functions on an open connected domain. For example there are Fréchet Montel spaces which are not Schwartz, Fréchet Schwartz spaces without approximation property, non-distinguished Fréchet spaces and Fréchet spaces with a continuous norm such that their bidual is isomorphic to a countable product of Banach spaces, among many others. See more information about these examples in [4] and [13].

Acknowledgement. This research was partially supported by the projects MTM2016- 76647-P and GV Prometeo/2017/102.

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Author’s address:

JoséBonet: Instituto Universitario de Matemática Pura y Aplicada IUMPA, Univer- sitat Politècnica de València, E-46071 Valencia, Spain email: [email protected]